Let S be a set of integers and denote the characteristic function of S as $\chi_{S}(n)$. Define an operator on the space of trig functions by the relation $\hat{Tf}(n) = \chi_{S}(n) \hat{f}(n)$. Here $\hat{f}(n)$ is the n-th Fourier coefficient of f.

For $p\geq 2$ we'll call S a $L^p$ multiplier set (or just $L^p$ multiplier) if there is an inequality of the form $\Vert Tf\Vert\_{p} \leq c \Vert f\Vert\_p$. If this inequality holds for some p but fails for $p+\epsilon$ for every $\epsilon>0$, we'll say that S is a strict $L^p$ multiplier.

Note that every set is a $L^2$ multiplier and if S is a $L^p$ multiplier for some p then it is a $L^q$ multiplier for $2 \leq q \leq p$. Moreover, it follows from a result of Zygmund that almost every (in the obvious sense) set is a strict $L^2$ multiplier. (I also think you can prove this via Khintchine's inequality, but I haven't checked this argument.)

Do strict $L^p$ multiplier sets exist for every $p>2$? Note that this is similar to the $\Lambda(p)$ problem, however, I don't see how to transform a strict $\Lambda(p)$ set into a strict $L^p$ multiplier set.

Thanks! It turns out that the proof follows easily from Bourgain's \Lambda(p) set construction. Let E be a strict \Lambda(p) set. Clearly E is a is a L^{p} multiplier, since ||Tf||_{p} << ||f||_{2} << ||f||_{p}. Conversely, if E was a L^{p+\epsilon} multiplier, we can show it must be a \Labmda(p+\epsilon') set by interpolating the estimates ||Tf||_{p} << ||f||_{2} with ||Tf||_{p+\epsilon} << ||f||_{p+\epsilon}. It was this last step that I was missing.
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Mark LewkoOct 31 '09 at 3:25